jackiw r. quantum meaning of classical field theory

We discuss how solutions to field equations, treated as classical, c-number nonlinear differential equations, expose unexpected states in the quantal Hilbert space with novel quantum nu

H Jackiw

Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139

Recent researches have shown that it is possible to obtain information about the physical content of

nontrivial quantum field theories by semiclassical methods This article reviews some of these

investigations We discuss how solutions to field equations, treated as classical, c-number nonlinear

differential equations, expose unexpected states in the quantal Hilbert space with novel quantum numbers

which arise from topological properties of the classical field’ configuration or from the mixing of internal

and space-time symmetries Also imaginary-time, c-number solutions are reviewed It is shown that they

provide nonperturbative information about the vacuum sector of the quantum theory

CONTENTS

B Quantum meaning of static, c-number fields 684

C Quantization about static, c-number fields 688

D Time-dependent, c-number fields 690

E Quantum meaning of time-dependent, c-number

F Quantization about time-dependent, c-number

III Models in Three Spatial Dimension 694

B Quantum meaning of static, c~number fields 695

C Quantization about static, c-number fields 696

E Fermions from bosons, spin from isospin 697

G Quantum meaning of imaginary-time, c-number

If quantum field theory is to be effective in describing

physical processes, as an emerging consensus among

theoretical physicists indicates, we must learn how to

perform accurate but approximate calculations, since,

due to their complexity, the relevant equations cannot

be solved exactly One approximation scheme, pertur-

bation theory, has been extensively developed and is

marvelously successful in some contexts, especially in

lepton-quantum electrodynamics Yet it is also clear

that there are phenomena which can never be seen in

the ordinary perturbative expansion For example,

there may be no small parameter in which to expand;

or even if such a small parameter exists, it may be that

the phenomenon we are studying is not described by

formulas which can be expanded in that small param-

eter—such an expansion may be singular In the last

*This work is supported in part through funds provided by

ERDA under Contract EY-76-C-02-3069,

Reviews of Modern Physics, Vol 49, No 3, July 1977

two years new approximation techniques have been de- veloped for calculating in quantum field theory, which avoid some of the shortcomings of the perturbative ex- pansion These nonperturbative calculations have ex- posed an unexpectedly rich particle structure in the quantal Hilbert space; they have put into evidence novel effects, like emergence of fermions from bosons; and they have provided new mechanisms for spontaneous symmetry breaking without Goldstone bosons Although the practical significance of all this for describing the present experimental data is obscure, we have clearly learned that a quantum field theory gives rise to phe- nomena of a much richer variety than had been believed heretofore

This article reviews some of the relevant investiga- tions Since the topic is large and supports many dif- ferent approaches, I shall mostly limit the discussion

to research done in Cambridge (U K and U.S A.) In all investigations, one begins with the first approxima- tion in which quantal effects are ignored, and treats all equations as if they were describing classical field con- figurations, rather than quantum operator fields Quan- tum mechanics is regained by quantizing the classical solution through semiclassical, Wentzel-Kramers-

Brillouin (WKB) methods (Dashen, Hasslacher, and

Neveu, 1974a,b; Korepin and Faddeev, 1975) Alter- natively, the full quantum theory can be expanded in the Born—Oppenheimer fashion, for which the first term is computed classically, and quantum corrections are found in a series expansion (Goldstone and Jackiw, 1975), It is this second, systematic approximation that

will be discussed here,!

In the first approximation we treat the Heisenberg operator field equations as c-number field equations and analyze them by methods of mathematical physics Classical solutions may be categorized as follows: con- stant solutions (time—and space—independent); static solutions (time-independent but space dependent); time- and space-dependent solutions; and lastly solu- tions to modified field equations, where the modification consists of replacing the time variable by an imaginary- time variable (f+ — ix,) The quantal significance of the

‘For previous alternate reviews see Jackiw, 1975; Rajara- man, 1975; Coleman, 1975c; Gervais and Neveu, 1976; Arno—

witt and Nath, 1976

constant solutions is known: they are first approxima-

tions to the vacuum expectation value of the quantum

field and frequently signal spontaneous symmetry viola-

tion, i.e., the Goldstone phenomenon The quantal

meaning of static and time-varying solutions will be ex-

plained in Sec II and III, first in the simple context of

models in one spatial dimension, and then for realistic

models in three spatial dimensions It will be demon-

strated that these classical solutions signal the presence

of particle states which had not been previously seen in

perturbative analyses The imaginary time solutions,

also analyzed in Sec III, are evidence for quantum me-

chanical tunnelling-~another nonperturbative effect

il MODELS IN ONE SPATIAL DIMENSION

In order to encounter first in a simple setting the

ideas that I wish to review, let us consider a quantum

field theory of a spinless field (x ,#) in one spatial

dimension The Lagrange density is assumed to be of

the form

For the energy to be positive definite, we take the

field potential U(#) to be non-negative

U(®) will in general depend on various numerical param-

eters (coupling constants) We wish to have a unique

parameter gfor systematic expansions, hence we assume

that U(’) depends on g in a Scaled fashion

1

The operator equation satisfied by ® is

oe u@)= (2 + (9) Vn — ay5) P+ U(®)=0, 8 u'(®)=0 (2.3) -

(The prime denotes differentiation with respect to argu-

ment.) I shall discuss first static, c-number solutions to

this field equation, and then their quantal meaning

Time-dependent solutions will be analyzed only briefly

A Static, c-number fields

Let us for the moment ignore the quantal nature of

Eq (2.3) and seek its static solutions The c-number

field y(x,t) satisfies

which for static configurations @(x) reduees to

The solutions which are of interest to us are delimited

by two requirements Firstly we demand that if ý,

solves (2.5), then £,(y,) should be finite, where # (2)

is the energy of a static field configuration 7

Eu(0)= Í ax {3 (o'+ UP}

The reason for this requirement is that E,(¢~,) will be

identified with an approximation to the energy eigen-

value of a new state in the quantum theory, and obvious-

ly this should not be infinite, Secondly we demand clas-

(2.6)

Sical stability Classical stability may be discussed in two different ways First we may return to the full, time-dependent equation (2.4) and write a time-depen- dent solution as

glx ,t)= oo(x)+ o,(*) expiw,t) (2.7)

where ¢, is our time-independent solution and the other term is a time-dependent perturbation, labeled by a parameter k Substituting this Ansatz into the full time- dependent equation (2.4) and linearizing around the small perturbation ¥, gives a Schrédinger-like equation for »,

d?

[ - Fa + 16 p.be))] tebe) = ody 60) (2.8)

Classical stability will ensue when the eigenvalues w? are non-negative, so that small perturbations about 7,

do not grow exponentially intime The other way of for- mulating classical stability is a variational one Equa- tion (2.5) can be obtained by demanding that the energy functional E,(¢), given in Eq (2.6), be stationary with respect to variations of ¢

0=6£,(¢)/5¢(x) =~" + U'(¢) (2.9) For stability it is required that the second variation

OE (PY) _ | Bam |

5jŒ)ö@) ` LT a5 tƯ (2)| 86 —?)

evaluated at the solution ¢,, be a non-negative differen- tial operator Clearly this means that all eigenfrequen- cies of (2.10) at g=y%, must be non-negative, which again leads to a study of (2.8) So classical stability, in either formulation, demands that the eigenvalues of the Schrédinger equation (2.8) be non-negative The demand for stability is motivated by the requirement that the corresponding quantum state be stable

There are some things which can be said about our problem independently of the explicit form of U(¢), and a very important statement is the following: The Schrédinger equation (2.8) always has a zero-frequency solution, called the “translation mode.” For stability this must be the lowest mode To prove the existence

of the translation mode, we differentiate (2.5) and ob- tain [note that U(v~) does not depend explicitly on the position x]

Moreover, this is a normalizable solution since

f dx(p))?<, because the energy is finite An equiva- lent way to understand the occurrence of a zero-fre- quency mode is the following: Translation invariance assures that if ¢,(x) is a solution of (2.5), then 9,(x + x9) will also be a solution Expanding the latter 9,(x+x,)

= ~, (x) + x, ¢i{x) and comparing with the expansion (2.7)

@(x, f)= œ() + j;„(x) exp@w,t) we see that g! isa small fluc- tuation and the frequency of this fluctuation is zero

It should be clear that the above generalizes to more

than one dimension—if ¢,(x) is a static solution, then the translation mode V¢,(x) is a zero-frequency “small-

oscillation” mode Also symmetries other than the

translation symmetry give rise to zero-frequency

modes If the energy functional is invariant under a

transformation g~> ¢+59, then 59, will be a zero-fre-

quency mode,

We may now prove another general result: Only in

one spatial dimension is it possible to find stable,

static solutions of finite energy in a spinless model

governed by the simple Lagrangian (2.1).? In more than

one dimension one must necessarily deal with more

complicated models, for example models with spin

This result becomes self-evident when it is realized

that the zero-frequency, translation mode ind dimen-

sions is d-fold degenerate (it is a p wave); on the other

hand, for the ordinary Schrédinger equation the lowest

eigenstate is nondegenerate Hence for d>1 there must

exist solutions to (2.8) with wZ<0 The same conclusion

can be obtained by a scaling argument Static field solu~

tions ¢,(x) in d dimensions stationarize the static ener-

gy E,(v)= f dx{i(Vev)?+ U(e)} If (x) is such a solution,

and y,(x)= v,(x/a), where a is a positive parameter,

then £,(¢,) must be stationary at a=1 A change of inte-

gration variable shows that E,(9,)=a?"E7(,)+ @E,(9,),

But now it follows that

9%E (0) seal | =3(3~4)Ez(Ø,),

ast

so that £,(?,) will be minimized only for 2-d>0O, i.e.,

d= 1

It should be emphasized that this negative result ap~

plies only to static solutions of the simple model (2.1)

In more than one dimension, it is easy to find static,

stable, finite-energy solutions provided the model in-

cludes Yang-Mills fields —this will be discussed in

Sec Il The simple one-dimensional model remains

an interesting laboratory for our theoretical ideas be-

cause ali the problems of developing a quantum theory

around a classical solution can be posed and answered

Moreover, the method of quantization carries over to

higher dimensions, and will be employed in Sec III for

the Yang-Mills models,

We return therefore to the one-dimensional Eq (2.5),

whose first integral may be given for arbitrary U

An integration constant does not appear in (2.14) so that

the energy will be finite [Equation (2.14) is consistent

“This theorem is well known to investigators of nonlinear

field equations An early reference which emphasizes the

relevance to particle physics is Hobart, 1963

Rev Mod Phys., Vol 49, No 3, July 1977

with the virial theorem (2.13) at d=1: E,=E,.] To inte- grate (2.14), we need an expression for U(2) There are many formulas for U(¢v) which lead to static, stable solutions with finite energy I shall discuss two exam- ples explicitly; however, our theory is independent of the specific form of U(¢)

U(o)= "Z| 1 - cos mes (2.15b)

Note that both theories possess discrete symmetries

(2.16a) (2.16b)

but the minima of the potentials, U’(¢y,)=0 [these are

constant solutions to (2.4)]

g* theory; g-~ 9

SG theory; @~~+@+ 2nwu(w/⁄g); n=0,+1,

(2,17a) (2.17b)

gy’ theory; ?,=+m/g,

SG theory; ¢,=27n(m/g);n=0,+1, ,

indicate that the symmetries are spontaneously broken

by the vacuum state Thus, according to the usual pro- cedure, we assign to the quantum field ® the vacuum expectation values

(2.18a) (2.18b)

gy? theory; (0|@/0)=m/g,

SG theory; (0|#|0)=0,

and upon expanding (2) about the vacuum value of ¢ we learn that the mass of the “mesons” in the ¢* theory is 2m, while in the SG theory it is m

Position-dependent solutions to (2.14) are the follow- ing:

g* theory; p(x) =+(m/g) tanhm(x —x,), (2.19a)

SG theory; 9,(x)=+4(m/g)tan“exp(+m(x—x,)) (2.19b) The occurrence of the parameter x, is a consequence of translation invariance; frequently we shall set it to zero The classical energy of the solution is finite

with very simple properties One finds a continuous

spectrum for w? =k? + L?, k? > Owith /„(2) ~ exp(ékz) multi-

plied by a Jacobi polynomial of degree LZ in tanh z.

(There is no reflection, only transmission.) In addi-

tion w? takes the discrete values L*-n’, n=L,

L-1, ,1 For (2.21) this means that in the y* theory

there is a zero-frequency solution ¢/, a second dis-

crete eigenvalue, and a continuum beginning at w?

= (23)?; in the SG theory the zero-frequency state is

again ~/, and the continuum begins at wj=m’ The

eigenvalues are non-negative; both solutions are stable

Note that in both cases the continuum begins as “? where

i is the mass of the meson

Let us observe that the static solutions (2.19) are

O(g), just as are the constant solutions (2.17) They

interpolate between the constant solutions as x ranges

from —° to ©, Also the energy density & =2(¢/)?+ U(¢,)

=(¢/)? is localized at x=x, We shall call solutions that

have a localized energy density for all time “solitons.” *

The soliton, though arising in a classical field theory,

looks very much like a classical particle, Its energy

density is localized at a point, its total energy is finite,

and itis stable Moreover, because the field equations are

Lorentz invariant, once we have the solution ¢,(x), we

also have the boosted solution 9, (x —vi/V1— v?) for

arbitrary v, |v|<1 The soliton can move in space

This then completes our discussion of static solutions

in the simple examples We now turn to the question of

the quantal significance of such solutions,

B Quantum meaning of static, c-number fields

In order to fit the static c-number solutions into a

quantum theory, we shall posit postulates about the Hil-

bert space of states, and we shall verify self~consis-

tently the validity of the postulates Also a systematic

expansion scheme will emerge, with which one can com-

pute quantal amplitudes to arbitrary accuracy

We postulate the existence of a vacuum state |0),

which in our examples is degenerate A particle space

is built on one of the vacua—there is no tunnelling,

hence we need not concern ourselves with the Hilbert

space built upon the other vacua There are of course

“meson” states, The one-meson state |k) describes a

stable, spinless boson with momentum Fk and mass /#,

and there are also multi-meson states |k,,k,, ) We

call this the “vacuum” sector and calculations in the

vacuum sector are performed in the standard way: The

quantum field @ is shifted by the constant solution ¢,

(m/g in the ¢* theory, 0 in the SG theory)

and conventional perturbation theory may be used to cal-

culate amplitudes of © Note that ¢, (when nonvanish-

ing) is O(g™); it is the lowest-order (in g) approximation

to (0|@|0) Multi-meson amplitudes involving & are of

higher order ing

The above description of the quantum theory was, un-

‘The nomenclature, advocated by T D Lee, is a borrowing

from the literature of applied mathematics and engineering

In those disciplines, however, “soliton” is used in a more

restrictive sense For a review of the older mathematical-

engineering researches see Scott, Chu, and McLaughlin,

1973; Whitham, 1974

til recently, traditional, but it is incomplete since it does not take into account the static c-number solutions (Goldstone and Jackiw, 1975) In order that these solu- tions be properly included, we further postulate that, in addition to the meson states, there exist other particle states, the quantum soliton states The one-soliton states | P) are momentum and energy eigenstates

P|P)=P|P), H|P)=£(P)|P),

In addition there are of course one-soliton, multi-meson

states |P;k,,k,, ), where P is the total momentum, and the k; are the asymptotic meson momenta, Also multi-soliton states exist, but we shall not be discus- sing them We postulate that the soliton is absolutely stable against decay into mesons; this means that all matrix elements of the form

(soliton, meson state | ® ,|no-soliton, meson state) vanish identically This sector of the Hilbert space is called the “soliton” sector

Next it must be decided whether there is only one type

of soliton, or whether there is a variety To settle this

we look at the variety of available static, c-number solutions with the same energy, which, as will be pres- ently demonstrated, are relevant to the soliton sector (just as the constant, c-number solutions are relevant

to the vacuum sector) Always there is a variety cor- responding to the symmetries of the problem In the examples considered, this is the variety labeled by x, arising from translational symmetry, as well as the variety of the sign of the solution corresponding to the field reflection symmetry; moreover, in the SG theory there is the variety of the inverse tangent’s multiple branches, arising from the discrete field translation symmetry Such symmetry-related varieties are of no consequence for distinguishing different types of solitons

If, however, there isa further variety to the classical solution, then we postulate that there are as many dif~ ferent types of solitons as there are varieties of static, c-number solutions with the same energy Thus in the y* theory, there is only one soliton; in the SG theory there are two, corresponding to the + variety of the ex- ponential We may call them soliton and anti-soliton; in what follows we shall concentrate on the soliton sector, corresponding to +, with the understanding that there is also an anti-soliton sector which does not communicate with the soliton sector

The last set of postulates sets the magnitudes of ma- trix elements of the quantum field @ in the soliton sec- tor We shall show self-consistently the following to be true:

CP! Ry yee Rip |®|P;h,, ; bưc = O(g 9051)

[Here and subsequently, it is understood that the field operator when written without argument is evaluated at

(2.26)

the origin, viz (®) =((0)).] The subscript C denotes

the connected part, where only the mesons are discon-

nected Thus according to the above

(P’| | P)=O(g"),

Œ?|®|P;k)= O(g9)

(P';k' | &| Pk) = (21) 5(k’ ~ kX P’|&|P)+(P’; kh’ | O|Pi Bo,

To show that our postulates can be verified self-con-

sistently, we begin by considering the one-soliton ma-

trix element of ® We know that the quantum field satis-

fies the operator equation

(2.28) (For definiteness we discuss the ¢* theory, but the

method is general) Let us take matrix elements of this

equation between soliton states

1[E(ŒP') - E(P)]? —[P! - P]?+ 3m°} ƒ(P',P)

= 2g?(P' |®3|P) = 2g°2_ ŒP?|®|z)@| ®|n@u | ®|P),

(2.29) where the field form factor ƒ has been defined by

As a consequence of the soliton’s stability, only states

in the one-soliton sector contribute to the completeness

sum in (2.29)

Equation (2.29) is exact; we now analyze to lowest or-

der ing The left-hand side is O(g™), since according

to our postulate that is the magnitude of f, and the re-

maining factors are O(g°), On the right-hand side the

factor g* requires that only terms of O(g7%) be kept in

the sum, But these can arise only from one-soliton

intermediate states Hence to O(g™) we may replace

(2.29) by an integral equation for f

{[E(P’) - E(P)]? -[P’ ~ P]?+ 2m?}¢(P’, P)

ÄP"äP” /',P")ƒ(P*,P*)ƒ(",P)

Upon comparing the g-behavior of both sides of the equa-

tion one concludes that f(P’,P) is, indeed, self-consis-

tently O(g™') In fact the equation may be simplified,

and solved completely to O(g"!)

The further simplifications are effected when it is re-

called that E(P’) - E(P)= VP"?+ M* —- VP?+M" Since by

hypothesis M is O(g™?) the energy difference when

expanded in powers of g is O(g”) and may be

dropped in a lowest-order O(g™') calculation The

physical meaning is that the soliton is very heavy

for weak coupling Therefore to leading order it

does not move, and its energy is just its rest mass, M

Next, observe that the function f(P’,P) depends only on

the difference P’~P, to leading order ing To see this,

recall that Lorentz invariance insures that f can be

only a function of the Lorentz scalars (P’, —P,,)? and

e""P P) To leading order (P/ -P,,)?=[E(P’) - E(P)]?

gi theory; (P’||P)= f dx exp[i(P’ ~ P)](m /g) tanhmx

+ higher powers in ¢ (2.36a) (The plus sign is taken in analogy with the vacuum sec-

tor, (0||0)=m/g+ higher orders ing The minus sign

is relevant to a parallel Hilbert space of identical struc- ture Also the arbitrary origin, x), is physically unin-

teresting; it gives only an arbitrary phase to (P’|®|P),

which we henceforth set to zero.) A similar calculation for the SG theory also determines the field matrix ele- ment:

SG theory; (P’|®|P)= ƒ dx exp[i(P’ ~ P)x](4m/g) tante”*

+higher powers in g (2.36b) This is the matrix element between soliton states; there is alsoa matrix element between anti-soliton states witha negative exponential; see (2.19b) There is no transi- tion matrix element between the soliton and anti-soliton states,

Consequently a first result has been obtained: a clas- sical, static solution can be fitted into the quantum theo-

ry provided we allow for new states—the soliton states Then, in an expansion in powers of g, the Fourier trans- form of the classical solution is the first approximation

to the field form factor (P’|#|P) which is O(g"),

The next thing that has to be checked is whether the soliton’s mass is, as postulated, of order g™*;, indeed

we have to determine what the mass is Let us compute the energy

5 =$h244(b')24 UB), (2.37b)

Hence

E(P)=(P |3¢|P)= VP 74M", (2.37e)

We expand /#(P) in powers of ø The first two

terms are M+ P?/2M SoM*(P |e? 4 2(®')?4+ U(6)|P),

where only the dominant, O(g7?) terms are kept Let us

begin with the evaluation of (P| ?|P)=27,{P|6|n)<n||P)

We need to compute this matrix element to order g”*;

therefore only the one-soliton states need be taken into

account in the intermediate state sum However, there

is a time derivative which is equivalent to the energy

difference But to leading order, the energy difference

between soliton states is zero, because to leading order

the energy is just the mass M So (P|?|P) may be

dropped to O(g") Next consider (P| (®’)?| P)

=2)<P |’ |nXn|o’|P) Again since we are computing

this quantity to order g~?, we need to keep only single-

soliton states (P| (#’)?|P)= [dP’ /(2rP|®’| PP’ | 6’ | P)

= [ dP’ /(2)(P'-P)XP|o |P?24P/|®|P) Substituting into

this the expression for (P| |P’) in terms of o(x), Eqs

(2.30) and (2.33), and carrying out all the integrations,

ene obtains (P| (#’)?|P)= J dx(o’(«))? Finally taking ma-

trix elements of U(®) and keeping only the single-soli-

ton states in the intermediate states, which is all that is

needed to O(g7"), one arrives at a formula for M in

terms of @(x):M= { dx[3(’)?+ U(b)] Since @ has been

shown to coincide with ¢,, we obtain:

We have thereby verified the self-consistency of the

postulate (2.25) that the mass of the soliton is of order

ge” Also we find that /@ coincides with the classical

energy in lowest order Of course, there are correc-

tions of higher order ing

The above calculations show that to lowest order in

the coupling constant our postulates about the soliton

sector are consistent Various quantal objects can be

computed; they are related to corresponding classical

quantities Moreover, we see that some quantum struc-

tures in the soliton sector (the field form factor, the

soliton mass) are proportional to inverse powers of

g—they are singular at g=0 and cannot be seen in ordi-

nary perturbation theory Nevertheless these irregular

contributions can be isolated and completely calculated

by the methods here developed Moreover, corrections

of higher order in g can be systematically computed

We now give an example of such higher computations;

we calculate everything to next order ing As we shall

see, an important quantum consistency condition emer-

ges which establishes the Poincaré covariance of our

method,

To evaluate first-order corrections we may still keep

E(P) independent of P, since the kinematical dependence

enters in O(g”), two orders beyond the lowest O(g7?)

However, in the saturation by intermediate states we

must keep the one-soliton one-~meson state |P; k); an

expression for (P’|®|P;)=/,(P’,P) is needed The

exact equation for that quantity is (in the ¢* theory)

{[E(P’) ~ E,(P)]? -[P’-P]?+ 2m} F,(P’, P) = 2g%P’ | &°| P)

(2,39)

Here £,(P) is the energy of the one-soliton, one-meson state; P is the total momentum; & is the meson momen-

tum To lowest order we take E,(P) to be M,+, In

saturating the right-hand side we keep the no-meson and the one-meson states, thus encountering the follow-

ing matrix elements: (P’|®|P),(P’|®|P3;k) and

(P';k' | ®|P;k) The first is known to lowest order; the second is being calculated; the third we decompose into

a connected and disconnected piece, as in (2.27) To the order we are computing only the disconnected piece is kept Also we take f,(P’,P) to be, in lowest order, a function of P’—P, the total momentum difference With these simplifications (2.39) becomes

Since (x)= y,(x) in lowest order, (2.42a) becomes the Schr6dinger equation (2.8):

2

| E, + uw.) (83x) = wells) (2.42)

Now we have a clear physical interpretation for the solutions of this equation The continuum solutions, which begin at w,= vk*+ u?, where u is the meson mass, are interpreted as meson—soliton scattering states If there are discrete states, other than the zero-frequency state (as in g* theory), they are excited states of the soliton The zero-frequency solution of (2.42) is not associated with any state For later convenience let us set $(;x) equal to (1/v2w, )u,(x) for all states with the exception of the zero-frequency state The normalized zero-frequency state is g{(x)/VM,, since M,= fdx(gl)*

Is it consistent to exclude the zero-frequency mode; i.e., are the physical states complete even though we are excluding one of the functions which contribute to a set of mathematically complete functions? Note also that (2.42) does not determine the normalization of p,(x) To settle both these points, we consider matrix elements of the canonical commutator between one-soli- ton states

(P’ |[@@&, 0), &(y, 0)]|P) =46 (x ~y)(27)6(P’-P)

(2.43)

Upon saturating with no-meson states and one-meson

states, the contribution of the one-meson states can be

shown to be

id, si fac exp [—i(P’- P)z]

be -2) wy -2)

The prime on the sum indicates that the zero-frequency

state is excluded, If we take the ¥,’s to be properly

continuum normalized, then the sum can be evaluated

by completeness A delta function does not emerge, ˆ`

since the zero-frequency mode is excluded; rather we

get

(2.44a)

i5(« —y)(27)5(P’ —P) -if dz exp| -i(P’- P)z]

x Cer -2) gly -z) - a

Next the no-meson contribution is evaluated; here are

encountered contributions of the form

are each O(g); consequently we must retain E(P”)

~ E(P) to order g’, since we are computing the commu-

tator which is Ó(ø°), The energy difference is taken to

be (P”? — P?)/2M', where we have put a prime to dis-

tinguish the mass that occurs in the kinetic term from

the rest mass; we shall prove that in fact M’=M, Eval-

uating the relevant integrals gives the no-meson contri-

bution to O(g°):

T fa exp[ - ¿P'—.P3z]#¿(x —z)@/(%w —z} (2.44d)

Thus when M’=M,, (2.44d) cancels the second term in

(2.44b), and the canonical commutation relation, the

hallmark of quantum mechanics, is regained

By this exercise we have learned three things First,

the properly normalized matrix element is (P’|®|P;k)

= fdx exp[i(P’ — P)x][¥,(x)/V2a, ] where ¥,(x) is a nor-

malized solution of the Schrédinger equation, hence

O(g°), consistent with the postulates (2.26) and (2.27)

Second, the zero-frequency solution is not a state of the

theory, rather it describes the first correction to the

motion of the soliton, Third, to the order computed, the

theory is Poincaré invariant since the rest mass coin-

cides with the kinetic mass

From the scattering solutions of (2.42) the meson—

soliton S matrix can be found For the ¢* and SG theo-

ries there is no reflection, only transmission The

transmission amplitude T is a pure phase by unitarity:

= exp|225(k)]

La tan5(k)= — +>

Rev Mod Phys., Vol 49, No 3, July 1977

Note that phase shift is independent of g

With the one-meson matrix element determined, the first-order correction to the energy and soliton field form factor can be computed Returning to (2,29) and retaining the one-meson states in the saturation of the right-hand side, we find that the equation satisfied by

1

329%

The soliton’s mass, through O(g°), is the classical en- ergy plus half the sum of the small fluctuation frequen- cies—a completely reasonable, quantum mechanical formula [Here again we See the need for non-negative eigenvalues in the Schrédinger equation (2.42): If wz

<0 then the soliton’s mass becomes complex—it is an

unstable particle.|

The equations (2.46) and (2.47) have to be renormal- ized, Firstly the (infinite) vacuum energy has to be re- moved from 32 p0 Moreover, the mass parameter in the theory m, has to be renormalized just as in the vacuum sector, (These are the only infinites of spin- less theories in one spatial dimension without derivative couplings.) The mass formulas have been renormalized and evaluated (Dashen, Hasslacher, and Neveu, 1974b; 1975) The results are:

Here ?# is the renormalized mass parameter

It is of course possible, with increasing tedium of computation, to extend the above method to the next, indeed to arbitrary order in g?.* Such computations have been performed; they provide an important verifi- cation of the consistency of our postulates about the soli-

“The study of a field theory through the equations satisfied

by matrix elements of the quantum field between states of a

very massive particle was pioneered in the context of nuclear

physics by Kerman and Klein, 1963 The method was applied

to the present problem by Goldstone and Jackiw, 1975 More recent developments are by Klein and Krejs, 1975, 1976; Klein, 1976; and Jacobs, 1976a.

ton sector (Jacobs, 1976a), I shall not review them

here, since there is available an alternate approach,

described in the following subsection, which gives a

diagrammatic depiction of the series in g

We turn next to the question of the soliton’s stability:

if it is heavy, why does it not decay into ordinary me-

sons? Stability is usually associated with an absolutely

conserved quantum number To see the existence of a

conservation law in our models, observe that

is a conserved current, not because it arises by No-

ether’s theorem from a symmetry of the theory, but

rather because it is trivially conserved, since it isa

divergence of an antisymmetric tensor The charge as-

sociated with this current is

In the vacuum sector the field tends to the same value

as x-~+°, and N vanishes In the soliton sector, the

field tends to different values, N is nonzero, and its

conservation renders the soliton stable (In higher di-

mensions, an antisymmetric tensor, whose divergence

is a conserved current, can be constructed with the help

of the spin degrees of freedom.) Such currents and con-

servation laws are called “topological,” since they arise

from topological properties of field configurations and

not directly from symmetries of the theory (Skyrme,

1961)

[It should be remarked that there exist localized,

finite-energy solutions whose stability arises not from

topological quantum numbers but from an ordinary, No-

ether conservation law These solutions are necessarily

time-dependent; I shall not discuss them here (Fried-

berg, Lee, and Sirlin, 1976).|

A final remark about the soliton: Observe that in the

vy? example #„(x) is an odd function of x, hence f(P’—P)

=(P'||P) will change sign when P’ and P are inter-

changed But by crossing symmetry an analytic contin-

uation of /(P’~P) should also describe the matrix ele-

ment (P’P||0) Antisymmetry of this matrix element

indicates that the solitons in the v* theory are fermions

(Goldstone and Jackiw, 1975) This truly remarkable

result—the emergence of fermions ina theory containing

only Bose fields—will be encountered again when we study

realistic three-dimensional models

C Quantization about static, c-number fields

Having established the existence of the soliton sector

and demonstrated the feasibility of a systematic coupling

constant expansion, it is appropriate to develop a dia-

grammatic perturbation theory, analogous to that in the

vacuum sector The approach there is to write =,

+; that is, the quantum field ® is shifted by y,, the

constant, O(g) solution to the field equations, and per-

turbation theory is developed in terms of the new field

& We would like to do something similar with the posi-

tion-dependent solution However, there are several

problems, First, ¢, depends not only on x but also on

Zo, the choice of the origin, Thus there are many clas-

sical solutions, parametrized by x,, and the question

is, by which classical solution should we shift? The second problem is that if we shift by a c-number which depends on x, then there will be great difficulty in im- plementing translation invariance because the mcmen- tum operator P commutes with the c-number solution

So, in order to maintain translation covariance, @ would have to transform in Some very complicated way The problem that we are facing is that we have a theo-

ry with a symmetry The symmetry is translational invariance But our classical solution is not symmetric,

a translation transforms it into another solution To overcome this problem and to develop a quantum theory around the classical solution, the procedure is to take the whole class of classical solutions parametrized by

x, and to promote x, in ¢,(x —x,), to a quantum variable X(t), This new quantum variable is called a “collective coordinate.” Therefore we write

In order not to increase the number of quantum degrees

of freedom we should set a subsidiary condition, and we take it to be

This condition is very convenient because ”} is a small oscillation associated with the zero-frequency mode which does not correspond to a physical state So the subsidiary condition insures that the quantum field

$(x,/) đoes not contain the unphysical zero-frequency mode,

To complete the specification of the canonical trans- formation we need to exhibit the transformation for the conjugate momenta The canonical momentum conjugate

to ® is i=6£/5b=6, The transformation conjugate to (2.51) involves a momentum P(/), conjugate to X(t), and

a field momentum I(x,t), conjugate to Š(x,/) The

E(t) = J dx 01 (xc) & (x,t)

Mo= ax(0¿(x)

As before we have to put a subsidiary condition on the canonical momenta so as not to increase the degrees of freedom

To verify that the above defines a canonical transforma -

tion, recall that the original variables satisfy i[II(x, 7)

&(\,4)]=5@ -y) Thisformula is regained, provided the

nonvanishing commutators of the new variables are taken

to be®

j_P0).XŒ)]=1

iil, 1), &(y OH] =5@ -y)- #(x)@¿(y) Mẹ : (2.55)

The physical interpretation of P() is that it is the to-

tal field momentum, which in terms of the old variables

is ~ ƒaxH(x,0)®'(,£) Upon expressing this in new vari-

ables and evaluating the integral with the help of the

orthogonality relations (2.52) and (2.54), we find P(t)

(Since the total momentum is a constant of motion, we

may drop the time dependence.)

Finally we exhibit the Hamiltonian in terms of the new

variables, which in old variables reads

H= ƒ dx1az1I?&, £)+ š(®! &,£))?+ U(® (, Ð)} (2.56)

Substituting (2.51) for ®, (2.53) for H, and shifting the

variable of integration from x tox+X, so that all fields

become evaluated at x rather thanx —~X, yields

H=M,+ P?/2M,+ HI, ®) - a (is £71 [ axtosy

Oo 0

(2.57a) where

P(l)= Mex f dx &' (x , tM («, £), itn, (2.57b)

(2.57c)

U(b, @,)= U(b+ v,) - 6" (y,) - U(@,) (2.574)

[The shift of integration of variable x ~x+X involves a

shift of a c-number by a g-number Hence one must

take into account the noncommutativity of X with P The

last term in (2.57a) arises from this quantum effect

(Tomboulis, 1975).] Note that H is independent of X,

hence commutes with P, which may be diagonalized and

taken to be a c-number The orders of magnitude in ø

are M,=O(g™),P,Il,@=0(g°), &/M,=O(g) Therefore

for a perturbative expansion, H may be separated in the

following way:

*Collective coordinates are widely used in many-body physics;

an early application to the polaron problem is by Bogoliubov

and Tyablikov, 1949; and to meson physics by Pais, 1957

For the soliton they were first discussed by Gervais and Sa-

kita, 1975; Callan and Gross, 1975; Korepin and Faddeev,

1975 These authors used a functional integral for the formu-

lation; however, the functional integral obscures problems of

quantum operator ordering The correct formalism, de-

veloped through operator canonical transformations, was given

by Christ and Lee, 1975; Tomboulis, 1975 For more recent

developments see Creutz, 1975; Tomboulis and Woo, 1976a;

Gervais and Jevicki, 1976a; Korepin and Faddeev, 1975;

Jevicki, 1976; Abbot, 1977

Rev Mod Phys., Vol 49, No 3, July 1977

H=Mạ+ Hạ+ Hị,

Hạ= 5 J dx[I1? + (&')? + U" (p,)6"] = O(ø9) (2.58)

Here H,=H —M,—-H, is the interaction part

When H, is ignored, H, can be diagonalized in terms

of the solutions of our Schrddinger equation (2.42), and

in this approximation Š can be written in terms of (me~ son) creation and annihilation operators

8x, = >) Fea logale) exp(— ial)

4 Wy,

+ az*(x) exp(26,/))

(The zero-frequency mode is not included in the sum since & is orthogonal to it.) Substituting this expansion

for & in H, we regain (2.47b) The soliton states are

labeled by P, the eigenvalue of momentum conjugate to A(#), which is also the total momentum One easily verifies that the field form factor is given to lowest or- der by the Fourier transform of ¢,, and other matrix elements of ® follow the postulate (2.26)

It is clear that a systematic perturbation series can now be developed, with 1/,+H, the unperturbed part and

H, the perturbation The perturbation theory is exactly the same as the one discussed in the previous subsec- tion, but now it can be represented by familiar graphical methods Various computations to high orders in g have been performed (Gervais, Jevicki, and Sakita, 1975; Jacobs, 1976a; Gervais and Jevicki, 1976a; de Vega, 1976)

Although the collective coordinate method for canon- ical quantization in the soliton sector has been presented for the simple example in one spatial dimension, it of course carries over to the three-dimensional theory as well Moreover, collective coordinates have to be intro- duced for all degeneracies of the problem, and the corresponding zero-frequency modes have to be re- moved by subsidiary conditions like (2.52) and (2.54) The soliton states will then be labeled by eigenvalues of the momenta conjugate to the collective coordinate, which commute with the Hamiltonian, if they generate symmetry transformations For example, in a model with charged fields 6,®*, or ina real basis $,,®,, if

there is a classical solution 9,= (%1), then because of

charge conservation one can obtain another solution by

a charge rotation

TẢ cos @ ~) (’ ) 6) (! sin cos@/ \vy,

In this case @ is the degeneracy parameter which de- scribes the symmetry—charge conservation, There will

be a zero-frequency mode 59, = (2) and the collective coordinate is O(¢) with conjugate momentum Q(/), which

in fact is time independent since it generates charge ro- tations The soliton energy eigenstates are also eigen- states of QM; they carry charge Since @ is conjugate

to an angular variable, it is quantized The energy wili depend on Q in order g*, and the dependence will be of

the form Q?/2 { dx(59,)? (Rajaraman and Weinberg,

The general expansion scheme here presented may be called a Born~Oppenheimer expansion for field theory,

(2.59)

since it is very analogous to the approximate calcula-

tions of the quantal properties of molecules In that con-

text also, the first approximation describes a particle

localized at an equilibrium point without kinetic energy,

while the second approximation exposes the harmonic

vibrational spectrum around the equilibrium It is only

in the next and higher approximations that rotational

degrees of freedom are encountered

D Time-dependent c-number fields

The previous subsections were devoted to an exhaus-

tive discussion of static solutions to classical field theo-

ry and of their quantum significance An interpretation

was given in terms of a new state in the theory —the

soliton—and the one-soliton sector of the Hilbert space

was analyzed It is plausible to suppose that time-de-

pendent solutions to the classical field equations of our

examples have something to do with multi-soliton states

As yet a complete and systematic perturbation theory

for the multi-soliton states has not been developed

However, using semiclassical methods for field theory

some results have been obtained We describe first the

types of time-dependent c-number solutions that one may

expect to find in a field theory

There are of course the entirely trivial time-depen-

dent solutions where a static solution has been boosted

This solution obviously continues to describe a soliton,

but now in a moving reference frame No new physical

information can be obtained, and we need not concern

ourselves with them

In some models with complex (charged) fields, it is

possible to find time-dependent, stable solutions with

localized energy density—soliton solutions—-even though

no such static solutions exist The time dependence is

in a phase, corresponding to nonvanishing charge It

has been shown that these solutions can also be associ-

ated with a stable quantum particle state—the stability

arises not from topological properties of the field con-

figuration, but from ordinary charge conservation

These very interesting nontopological solitons may have

something to do with the observed particles; but they

are outside the scope of this review and will not be dis-

cussed (Friedberg, Lee, and Sirlin, 1976)

The types of time-dependent solutions which are rele-

vant to the topological solitons under review here should

have the property that as t —%* they describe several

widely separated static solutions, moving towards each

other Clearly such solutions would be relevant to

multi-soliton scattering Also interesting are periodic

solutions that could be interpreted as multi-soliton

bound states Unfortunately, even in one spatial dimen-

sion the nonlinear, partial differential equations are

sufficiently complicated so that no general discussion of

such solutions is at present available However, for the

SG theory it is possible to integrate the equations com-

pletely and all solutions are explicitly available Of

necessity, I confine the subsequent discussion to the SG

solutions

In the SG theory one can find the static soliton and an-

ti-soliton solution, given in Eq (2.19b) There are also

time-dependent N soliton solutions with the following

properties The N soliton solution depends on 2N pa-

rameters As /-—©, the solution becomes a superposi- tion of N one-soliton solutions and the 2N parameters

correspond to asymptotic velocities v‘’’ and positions

x§[¡=1, ,N] of the N solitons As t-+, the solu- tion again decomposes into a superposition of N one-sol- iton solutions The asymptotic final velocities are the same as the initial ones, the asymptotic positions differ from the initial ones by an amount that can be ascribed

to a time delay in the multi-soliton collision By trans- lation invariance, two constants of motion can be arbi- trarily set to zero, and a third can also be made to vanish if the calculation is performed in the center-of- mass frame For example, the two~soliton solution de- pends on one constant, u, the relative velocity of the two solitons The explicit form of the two-soliton solutions is:

The total momentum of each solution is zero; the energy

is 2M,y,M,=8m°/g?, Examination of the asymptotic

forms of the two solutions shows that in both cases there

is time delay

There is another solution, the soliton, anti-soliton bound state or “breather” which is periodic in time It

is obtained from (2.60b) by taking w=ia, @ real

4m tant 2 Sinmyrat

= —— tan

Ye g a coshmyx ’ y= (14 a2) 1/2,

(2.62) The energy is 2M,y There is no soliton, soliton bound State

E Quantum meaning of time-dependent, c-number fields Once we have in hand classical, time-dependent solu- tions for a field theory, which is a very rare circum- stance indeed, we can do something different from the Born—Oppenheimer approximation scheme that we have described in connection with the static solutions Rath-

er, following Bohr and Wigner, we can perform WKB approximations as in quantum mechanics

Let us recall the WKB approximation in quantum me- chanics with one degree of freedom When a particle

is moving periodically with energy E in a smooth poten- tial between two turning points g, and qg,, then the WKB quantization condition is

f * VE ~2V@)dq= („+ š}n = ƒ ”p (q)dq,

where p(q)= Vv 2E —2V(q) is the local momentum This

is valid for large n, and the WKB approximation gives the first ‘wo terms ina large ” expansion of the energy.

The semiclassical, Bohr quantization condition, based

on the correspondence principle, determines the first

term, um The second term 37, follows from the details

of motion in one degree of freedom To generalize to

many degrees of freedom we remain with the Bohr con-

dition [It is possible to generalize the full WKB approx-

imation to many degrees of freedom (Maslow, 1970;

Gutzwiller, 1971) and even to afield theory with infinite

degrees of freedom (Dashen, Hasslacher, and Neveu,

1974a,b); I shall not be reviewing these.]

To develop the generalization of the semiclassical

method, we first rewrite the Bohr condition as

[”»@da= [ ”arp(Đ4()=m,

where the variable of integration has been changed from

qto/, Note that /, —/, is the semiperiod of the motion

For N degrees of freedom, the generalization is

[ras p, (tq, (t)=nn

Clearly, for a field theory, with an infinite number of

degrees of freedom, the Bohr, semiclassical quantiza-

where J,(£) is the classical action for a periodic solu-

tion with semiperiod T and energy E We are instructed

by (2.63) to find classical periodic solutions, y,, then

to integrate the product of the canonical momentum with

~, Over a Semiperiod and equate this to m, thus achiev-

ing one quantization condition on the parameters of the

solution

In the SG problem there exists a periodic solu-

tion, Eq (2.62), depending on the parameter a

Hence to quantize it, we integrate Ily,= 2 over all

space, and then over the semiperiod -17/2mya<t

<n /&mya, Equating the integral to mm results in a quan-

tization of the parameter a Since the energy is also

expressed in terms of a, E=2M,(1+ a7)", this is equi-

valent to a quantization of the energy The result is

(Dashen, Hasslacher, and Neveu, 1975; Korepin and

Faddeev, 1975)

Eạ= 2M, sin-y—n,1#= 1,2, , S aM, cra =— (2,64) 64

So in the quantum field theory there are soliton, anti-

soliton bound states

Let us now perform a semiclassical analysis of scat-

tering Wigner has shown that twice the derivative of

the phase shift in the semiclassical approximation is the

time delay

Hence we can compute the phase shift by integrating the

time delay (2.61) with respect to energy

E

Eth The constant of integration, the phase shift at threshold, may be evaluated as follows Consider the classical ac- tion 7„(E) for a solution to the equations of motion, with energy —£, which passes from an initial configuration to

a final configuration in time T Since d/,(£)/dT=-—E, it

follows that

dl ,(E£) aT -gữz(E)+ =r)- (HE dE +) dE + T=T (2.67a)

That is, the time of flight can be expressed as an energy derivative Total time delay is equal to the time of flight in the presence of forces, less the time of flight

in the absence of forces, in the limit as T goes to infini-

ty But the time of flight in the absence of forces is given by (2.67a), where the term in parentheses on the

left-hand side may be written as p(Z)[x,(T) -—x,], with

p(B) being the relative momentum of the particles and x,,X, the initial, final position Thus the total time de- lay is

A/(E)= lim Up(E) +ET —p(E)[x,(7) —x,]), - (2.67b)

and the phase shift can be taken to be

is true that

Comparing (2.69) and (2.70) we find

which we call the semiclassical Levinson”s theorem

[The exact Levinson’s theorem is 5(E,,) — ö(%) = (r /2)0z,

where the factor of % is peculiar to one-dimensional mo-

ton, soliton channel and 1, = 81m?/g? for the soliton, anti-

soliton channel, we find the following phase shifts (Jackiw and Woo, 1975):

(2.73a) _ *r”m?” 16? f Inv 1? M,

5g (E)= ge + oe j dx 1-x T0 mm

"`"

m Jo 1—x E=(8z3/g?)(1 —ø2)ˆ1/2= M¿(1 — 2)^12, (2.73b)

There are several interesting properties of the result

Note that for weak coupling (smali g), the phase shifts

are large This means that the forces are large and that

the solitons are strongly interacting even for weak cou-

pling In fact there are three scales of interaction for

the theory when g is small The conventional meson—

meson interactions are weak, since they are proportion-

alto g The meson-—soliton interactions are of inter-

mediate strength—independent of g [compare (2.45)]

Finally the soliton—soliton forces are strong Another

interesting feature of (2.73) is that crossing symmetry

is satisfied in the sense that 5,, and 5,, are related by

the crossing relation (Coleman, 1975b)

F Quantization about time-dependent,

c-number fields

In order to calculate contributions to the bound-state

energies and phase shifts, beyond the lowest-order ones

presented in the previous subsection, a systematic

series must be constructed for expanding the quantum

theory around a time-dependent, c-number solution

Such expansions have been found by functional WKB

methods (Dashen, Hasslacher, and Neveu, 1975) or by

time-dependent canonical transformations (Christ and

Lee, 1975) The formalism requires explicit time-de-

pendent solutions; consequently its applicability is con-

fined to the SG theory where these are available More-

over, it is sufficiently complicated so that only the first

correction has been computed I shall not review the

theory, beyond quoting the results of the first-order

computations which, in the end, turn out to be very sim-

ple (Dashen, Hasslacher, and Neveu, 1975; Gervais

and Jevicki, 1976b; Lee and Gavrielides, 1976): Both

the bound-state energies and the phase shifts are given

by the same expressions as in the semiclassical limit,

provided they are expressed in terms of the first-order

soliton mass M = (8m3/g?) —(m/r), rather than the low-

est-order mass M,= (8m?/g") It has also been possible

to extend the WKB method to a calculation of the (expo-

nentially small) reflection coefficient in multi-soliton

scattering (Korepin, 1976)

The SG theory has continued to interest theorists, and

several exact results have been obtained It has been

shown that the SG theory is equivalent to the massive

Thirring model (Coleman, 1975a) The fermion fields

of the Thirring model describe particles, which are

identified with the SG solitons, and the fermion number

current yy"? is proportional to the topological current

e€""8,6, Thus the SG solitons also are fermions, just as

are the ¢* solitons Furthermore, it has been possible

to obtain the bound-state spectrum of the solitons-fer-

mions exactly, and it is found to agree with the WKB ap-

proximation (Luther, 1976),

These exact results validate the semiclassical ap-

proach Also they expose a fascinating duality: the

same physical reality has two equivalent descriptions,

one bosonic with fermions appearing as coherent bound

states; the other fermionic where the bosons are bound

states Unfortunately there is no indication at the

present time that these marvelous features of the SG

theory are also to be seen in realistic, three-dimen-

sional models

G Effects of Fermi fields The models discussed thus far involve only Bose fields We now wish to summarize the new effects that arise when Fermi fields are included We consider the Lagrange density

1 = 2m?b — 2¢7b3 —~GgWw , (2.75a)

In the absence of the Fermi fields, this is just the ¢4 model, with O(g) soliton solutions The Fermi field equation (2.75b) is linear in ¥, whose interaction with the Bose field is of the form g®, hence O(g°) when ® is

O(g) [We take G to be O(g°).] Also the reaction of

the Fermi fields on & is seen from (2.75a) to be O(g) Hence for weak coupling (small g), we may ignore the Fermi fields in the first O(g"') approximation, solve the pure Bose equation, and then solve the Dirac equa- tion in the external, static, c-number Bose potential, thus obtaining an O(g°) Dirac wave function In other words, for the extended theory (2.74), a systematic coupling constant expansion may be given in which the Fermi fields enter only at the first correction to the lowest approximation

Before describing the quantal significance of the c- number Dirac wave function, let us discuss the solutions

of the Dirac equation (2.75b), where the Fermi field operator is viewed as a c-number wave function and g®

is given by the static solution to the ¢* theory Because the potential is static, we may look for energy eigen- functions

la 1 - +8G?n tanh 7 Ox | (x)= eb), (2.76b)

where a@=y°y! ,g=y°, The equation possesses the usual positive and negative energy solutions These are re- lated by a fermion-number conjugation operation, which

in a representation where @ =o" and g =o' is given by o° However, in addition to these, there is a unique, nor- malizable, zero-energy solution

ơŸỦạ = Ủo

We shall demonstrate that the occurrence of the zero- energy mode in the Fermion system has profound conse- quences for the physical interpretation of the quantum theory Although the solution is here exhibited in a very specific model, in fact it is present in much more general situations It appears, although this has not been proven, that any Dirac equation in a topologically

(2.78)

interesting potential with a conjugation symmetry, pos-

sesses a normalizable, self-conjugate, zero-energy

solution (Jackiw and Rebbi, 1976a, 1976c; 't Hooft,

1976a,b; Jacobs, 1976b; Grossman, 1977)

The quantum theory described by the Lagrangian

(2.74) has the following structure There is of course

(P's|W'[P 2sp-)= J de expli(P’ — Py Ju glc)= 0(g")

(2.79d)

(P'+|®|P+;k)= f dx exp[i(P’ ~ P)x | 0(k3x)/V¥ 2a, |

the conventional vacuum sector with mesons, fermions,

‘and anti-fermions, there is also a soliton sector, and in

the absence of Fermi fields we have demonstrated the

existence of the one-soliton states |P) This sector is

modified by the presence of Fermi fields which possess

a zero-energy solution to the c-number Dirac equation

The zero-energy mode signals quantum mechanical

degeneracy, and we postulate that as a consequence the

soliton states are doublets | P+) The additional label,

+, describes a twofold degeneracy which is required by

the zero-energy fermion solution We call the + state

“soliton” and the — state “anti-soliton.”” (This bifurcation

has no relation to any soliton multiplicity which is al-

ready present in the purely bosonic theory, as in the SG

example We are here describing a new degeneracy,

which is a consequence of the fermions.) It must be

stressed that we do not take the viewpoint that the soli-

tons exist independently of the fermions, which then bind

to them with zero energy Rather we say that the soli-

ton is doubly degenerate The difference is that the

first viewpoint would lead to four states: the original

soliton, soliton plus fermion, soliton plus anti-fermion,

soliton plus fermion and anti-fermion; our interpreta-

tion involves only two states,

The consistency of this picture is demonstrated by the

same method as in the purely bosonic theories: we list

relevant states in the soliton sector, postulate orders of

magnitude for various field matrix elements, determine

equations for the field form factors from the operator

equations of motion (2.75), expand systematically in the

coupling constant to regain the c-number equations, and

finally normalize various solutions by quantum field

commutation and anticommutation relations Specifi-

cally we consider the states

|P +) solitonor anti-soliton, with energy VP?+ M7,

|P+;k) soliton or anti-soliton plus one meson, with en-

The field form factors are defined

(P'+|@|P x)= f dx exp[i(P’ ~ P)x]o(«) = O(g")

(2.79a) (P’~|¥|P+)= f ax exp[i(P’ — P)x Ju, (x) = O(øg°)

gy solution with eigenvalue ¢,;,v, is the conjugate of the normalized negative energy solution with eigenvalue

~€, The normalization is determined by evaluating

the equal-time Fermi field anti-commutator (ŒP/+|{w'œ ,0), w(+y, 0)}, LP +)=ö(x — y)(3z)6(P'—P)

(2.80)

A compact summary of the physical picture that is be- ing presented can be given by the following expansion of the quantum fields In order to account for the transla- tional degrees of freedom, a collective position coordi- nate is introduced, as explained in Sec IC

The 6’s create or annihilate fermions, while the d’s per- form the same task for the anti-fermions The a opera- tor however does not create or annihilate particles; it merely connects the soliton with the anti-soliton

a|P+)=|P-), a'|P -)=|P4),